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  "body": "\n## Definition\n\nLet $R$ be a commutative ring and let $f \\in R[x_1, \\ldots, x_n]$ be a [[polynomial in several variables|0CES91M79A]].\nThe total degree of a monomial $x_1^{a_1} \\cdots x_n^{a_n}$ is $a_1 + \\cdots + a_n$.\nLet $d \\ge 0$ be an integer.\nThe polynomial $f$ is *homogeneous of degree $d$* if every monomial that appears in $f$ with a nonzero coefficient has total degree $d$.\nThe zero polynomial is homogeneous of every degree.\n\n## Remarks\n\nExample: $f = x^2 y + 3 y z^2 - z^3$ in $\\Q[x, y, z]$ is homogeneous of degree $3$.\nCheck: the monomials $x^2 y$, $y z^2$ and $z^3$ have total degrees $2 + 1$, $1 + 2$ and $3$.\n\nNon-example: $g = x^2 + y$ is not homogeneous, because its monomials have total degrees $2$ and $1$.\n\nThe polynomial $f$ is homogeneous of degree $d$ if and only if $f(\\lambda x_1, \\ldots, \\lambda x_n) = \\lambda^d f(x_1, \\ldots, x_n)$ in $R[\\lambda, x_1, \\ldots, x_n]$.\nEvery polynomial is in a unique way a finite sum $f = f_0 + f_1 + \\cdots$ where each $f_d$ is homogeneous of degree $d$; these are the homogeneous components of $f$.\nA homogeneous polynomial is also called a form; forms of degree $1$, $2$ and $3$ are linear, quadratic and cubic forms.\nIn one variable, the nonzero homogeneous polynomials of degree $d$ are the $c x^d$ with $c \\ne 0$, whose [[degree|0DPF6WRZW6]] is $d$.\n",
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